Page 288 - A Course in Linear Algebra with Applications

272 Chapter Eight: Eigenvectors and Eigenvalues

We know that A has at least one eigenvalue c in C, with
associated eigenvector X say. Since X ^ 0, it is possible
n
to adjoin vectors to X to produce a basis of C , say X =
X\,X 2,..., X n\ here we have used 5.1.4. Next, recall that left
n
multiplication of the vectors of C by A gives rise to linear
n
operator T on C . With respect to the basis {Xi,... , X n } ,
the linear operator T will be represented by a matrix with the
special form

where A\ and A 2 are certain complex matrices, A\ having
n — 1 rows and columns. The reason for the special form
is that T{X\) = AX\ = cX\ since X\ is an eigenvalue of
A. Notice that the matrices A and B\ are similar since they
represent the same linear operator T; suppose that in fact
Bi = S^ASi where Si is an invertible n x n matrix.
Now by induction hypothesis there is an invertible matrix
62 with n — 1 rows and columns such that B^ = S^ 1 A\Si is
upper triangular. Write

s = Sl
{o 1)-

This is a product of invertible matrices, so it is invertible. An
easy matrix computation shows that S^^-AS equals

which equals

Replace Bi by I . 2 ) and multiply the matrices together
to get